Consistent Signal Subspace Estimation for Wide-Band Sensor Arrays Elio D

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Space Time MUSIC: Consistent Signal Subspace Estimation for Wide-band Sensor Arrays Elio D. Di Claudio, Raffaele Parisi, and Giovanni Jacovitti

Abstract—Wide-band Direction of Arrival (DOA) estimation Maximum Likelihood (ML) [1]–[3], MUSIC [4] and with sensor arrays is an essential task in sonar, radar, acoustics, Weighted Subspace Fitting (WSF) [5], [6] narrow-band biomedical and multimedia applications. Many state of the DOA estimators, or a coherent focusing stage [7]–[13]. art wide-band DOA estimators coherently process frequency binned array outputs by approximate Maximum Likelihood, • Time Difference of Arrival (TDOA) estimation, where Weighted Subspace Fitting or focusing techniques. This paper wavefronts are reconstructed by fitting estimates of the shows that bin signals obtained by filter-bank approaches do differential time delays among pairs of sensors [14]. not obey the finite rank narrow-band array model, because • Wide-band adaptive steered beamforming, optimized by spectral leakage and the change of the array response with a Minimum Variance (MV) [15] or ML criterion in the frequency within the bin create ghost sources dependent on the particular realization of the source process. Therefore, existing frequency domain [16]. DOA estimators based on binning cannot claim consistency even • Compressed Sensing (CS), where the sparsity of sources with the perfect knowledge of the array response. In this work, is enforced by a penalty functional in a quantized DOA a more realistic array model with a finite length of the sensor domain [5], [17]. impulse responses is assumed, which still has finite rank under The resolving capability of TDOA approaches is limited a space-time formulation. It is shown that signal subspaces at arbitrary frequencies can be consistently recovered under mild by sampling and windowing and by the presence of multiple conditions by applying MUSIC-type (ST-MUSIC) estimators to sources and colored noise [14]. the dominant eigenvectors of the wide-band space-time sensor Wide-band steered beamforming and CS furnish very robust cross-correlation matrix. A novel Maximum Likelihood based but inconsistent DOA estimates, respectively asymptotically ST-MUSIC subspace estimate is developed in order to recover biased and resolution limited. consistency. The number of sources active at each frequency are estimated by Information Theoretic Criteria. The sample Since parametric narrow-band DOA estimation is consistent ST-MUSIC subspaces can be fed to any subspace fitting DOA and can cope with multi-source, correlated noise and arbitrary estimator at single or multiple frequencies. Simulations confirm array geometries, there is a strong interest in extending its use that the new technique clearly outperforms binning approaches to the wide-band case through frequency binning. However at sufficiently high signal to noise ratio, when model mismatches the observation time must be sufficiently small to comply with exceed the noise floor. the assumption of stationarity of the sources and to provide an Index Terms—Direction finding, wide-band sensor arrays, adequate number of independent snapshots for the stable SCM space-time processing, signal subspace, coherent focusing, MU- estimation within each bin. These contrasting requirements SIC, Weighted Subspace Fitting, WAVES, AIC, BIC. enforce an unavoidable spread of the binning filters in the frequency domain [18]. I.INTRODUCTION The problem is sketched in Fig. 1 for a uniform linear array IDE-BAND signal processing with sensor arrays is (ULA). The spectral support of each noiseless plane wave W a relevant field of research in widespread remote source propagating at speed u and impinging from azimuth sensing applications. In particular, Direction of Arrival (DOA) θ, referred to broadside, is a Dirac wall based on the line estimation in sonar, multi-band radar, acoustic surveillance, kx = −ω sin (θ)/u in the plane formed by the wavenumber arXiv:1711.01631v2 [stat.AP] 1 Mar 2018 seismics, video conferencing and Ultra-Wide Band (UWB) kx and the angular frequency ω, whose mass is modulated by radio mostly involves signal bandwidths that largely violate the source spectrum [19]. the narrow-band assumption. In binning approaches, best alignments of DOA estimates Wide-band DOA estimation is usually tackled by four main from different bins are inferred [5], [8]. However, the DOA approaches: ambiguity caused by variations of the source spectrum and the steering vector [20] within each bin and the spectral leakage • Frequency binning [1], performed by decomposing the among adjacent bins due to window side-lobes cannot be wide-band signals into a set of narrow-band signals recovered, since the narrowband array model does not account (bins), each tuned to a different frequency. The spatial for them. In particular: information carried by a set of narrow-band spatial covariance matrices (SCMs) is combined into a unique • Spatially spread, uncalibrated ghost sources emerge from DOA estimate by either a multi-frequency extension of noise background for increasing signal to noise ratio (SNR) and/or sample size [12], [21], [22]. Manuscript received March 2, 2018; revised Month, Year. • Therefore, the finite rank structure of the narrow-band The authors are with the Department of Information Engineering, array model [20] does not hold within each bin. Electronics and Telecommunications, University of Rome “La Sapienza”, via Eudossiana 18, 00184 Rome, Italy. (e-mail: {elio.diclaudio, raf- • The model becomes unidentifiable and the detection of faele.parisi}@uniroma1.it, [email protected]) the source number [23] is an ill-posed problem. 2 SUBMITTED

approaches. However, such estimators require the knowledge of the number and the impulse responses of all sources and a very accurate coarse DOA estimation, which is at least un- practical. In [22], a MUSIC [20] type condition allowed to asymptotically recover finite rank narrow-band signal subspaces at arbitrary frequencies from the the wide-band dominant eigenvectors of the STCM (or any other statistic sharing a similar structure [29], [30]), rather than from filtered array data. Earlier versions of this generalized Space-Time MUSIC (ST-MUSIC) [22], [31] subspace estimator did not allow for a sound statistical estimation of the number of sources and Fig. 1. The spectral support of four far field noiseless and uncorrelated had a slowly decaying asymptotic bias, which hampered the sources impinging on an ULA in the half plane whose coordinates are the numerical stability of the DOA estimates, especially at extreme the wavenumber kx and the angular frequency ω > 0 [19]. The wavenumber (and therefore the DOA) uncertainties due to the finite bin width and to a SNR and with small samples, without the use of heuristic typical focusing process [8] are illustrated. regularization [32]. Therefore, a novel ST-MUSIC subspace estimator, sharing similarities with the Signal Subspace-MUSIC (SS-MUSIC) • signal-dependent The model mismatch generates a DOA [33], was derived from the ML formulation of an inversion bias. problem. The ML setting optimizes the statistical efficiency • For random sources, an excess DOA variance appears at and increases the robustness of the sample signal subspace high SNR [12] and may even hamper the convergence of at extreme SNR, thus allowing the development of subspace optimal DOA estimators [24], such as ML [3] and WSF rank estimators, based on Akaike [34] and Bayesian [23], [35] [5], [6], [25]. Information Theoretic Criteria (AIC and BIC respectively). This mismatch undermines the consistency of the existing A perturbative analysis of the ML ST-MUSIC signal sub- narrow-band DOA estimators even in the absence of noise used space estimate established its consistency and the optimal on frequency bins and is especially relevant for wide spatial weighting in the WSF sense [6]. Therefore, the sample ML apertures or when wide-band, short-time acquisitions are re- ST-MUSIC subspaces at different frequencies can replace the quired. It adds up to calibration and focusing mismatches, all inconsistent SCM counterparts in any narrow-band or wide- relevant at high SNR [24], [26]. band subspace based DOA estimator [5], [11]–[13] regardless Consistency guarantees that the limit DOA solution for of the spatial coherency of sources. vanishing noise or increasing the sample size is unique and Numerical simulations of the ML ST-MUSIC subspaces satisfies the underlying physical model, regardless of signal with WSF based DOA estimation schemes [6], [12] show a realizations. This is important for scientific experiments, array definite statistical performance improvement at high SNR of calibration, target tracking and nulling static interference in the ML ST-MUSIC subspaces over the classical counterparts, telecommunications. approaching the Cramer Rao DOA variance bound, at the cost Our goal is to obtain a consistent estimate of the signal of a slightly increased low SNR threshold in some cases, subspaces, asymptotically free from artifacts to replace the whose causes are discussed, and higher computation burden. SCM counterparts. With reference to Fig. 1, it is clear that After a notation Sect. II, the finite length wide-band array this task requires a joint parametric modeling in space and response model is analyzed in Sect. III for DOA estimation time. purposes. In Sect. IV the ST-MUSIC concepts are developed. In [27] high resolution, narrow-band SCMs were esti- In Sect. V, the ML ST-MUSIC subspace estimator is devel- mated by suppressing the spectral leakage by a bank of MV, oped and AIC and BIC rank estimators are deduced, based on distortion-less, delay and sum (DS) beamformers, tuned to the first order finite sample perturbation models of the STCM. the same frequency. This approach involves cross-temporal In Sect. VI the computational analysis is sketched. In Sect. moments through the space-time array covariance matrix VII the validity of the ML ST-MUSIC model is assessed by (STCM), which only requires time sampling of sensor outputs numerical simulations. Finally, conclusion is drawn in Sect. at a sufficient rate. The STCM exhibits fast statistical conver- VIII. gence and low bias even for short data records. However, the resulting Capon SCM estimator of [27] suffers from the small II.NOTATION sample issues of MV beamforming [28] and its structure was not analyzed for DOA estimation purposes. Throughout the paper matrices are indicated by boldface, In this work, starting from the finite rank property of the capital letters, vectors by boldface, lowercase letters. The wide-band source signature on the STCM, under the assump- transpose of matrix A is indicated by AT , the conjugate by tion of finite lengths of the source to sensor impulse responses A∗, the Hermitian transpose by AH . The pseudo-inverse of A † [27], an extension of the narrow-band DOA identifiability is A . IM is the square identity matrix of size M. The operator conditions [3], [20] shows that consistent wide-band DOA diag{A} creates a column vector with the main diagonal of estimation is theoretically feasible by space time ML or WSF A, diag{a} creates a diagonal matrix with the elements of DI CLAUDIO et al.: SPACE TIME MUSIC: CONSISTENT SIGNAL SUBSPACE ESTIMATION FOR WIDE-BAND SENSOR ARRAYS 3

vector a placed on its main diagonal. trace(A) is the trace common effective overall length Ld = max1≤m≤M Lm,d for of matrix A. det(A) is the determinant of A. A ⊗ B is all hm (θd) related to the generic d-th source. the Kronecker product between matrices A and B. δkl is the Each impulse response hm (θd), zero padded to Ld, is used Kronecker√ symbol, equal to one if k = l and zero elsewhere. to build a Toeplitz convolution matrix of size P ×(P +Ld −1) j = −1 is the imaginary unit. Sub-matrices are indexed h (θ ) 0 ··· 0  by MATLAB-like conventions [36]. For instance, A(:, k) is m d  0 hm (θd) ··· 0  the k−th column of matrix A and A(1 : m, 1 : n) is the Hm (θd) =   (5)  ············  upper left submatrix of A of size m × n. The operator ∗ 0 0 ··· hm (θd) indicates the temporal convolution of two signals. E {x} is the expected value of the random variable x and var (x) denotes for m = 1, 2,...,M. Finally, the STS (2) is compactly its variance. The covariance between the random variables x expressed as and y is indicated by cov (x, y) . The Frobenius norm [36] of D X A is indicated by |A|F . Sample quantities are denoted by a xST (n) = H (θd) sd (n) + v (n) (6) hat superscript (e.g., Aˆ ). d=1

where H (θd) stacks Hm (θd) for m = 1, ··· ,M and is the III.ARRAY MODEL multi-channel convolution matrix of the sampled d-th source A sensor array with M sensors is immersed in a wave- signal T ﬁeld and receives the signals sd(t) (d = 1, 2,...,D), radiated sd (n) = sd (n) sd (n − 1) ··· sd (n − P − Ld + 2) by D < M wide-band point sources, whose DOAs are (7) characterized by the generic coordinate vectors θd. Provided and the MP × 1 vector that the sensors and the propagating medium are linear, the T T T T signals xm(t) (m = 1, 2,...,M), received by the M sensors v (n) = v1 (n) v2 (n) ··· vM (n) (8) at time t, are modeled as collects the additive sensor noise samples D T X vm (n) = vm (n) vm (n − 1) ··· vm (n − P + 1) . xm (t) = hm (t; θd) ∗ sd (t) + vm (t) (1) d=1 (9) Equations (5) and (6) show that the rank of H (θd) is upper where hm (t; θd) is the impulse response from the d-th source bounded by P + Ld − 1 [27]. The SCM model [20] can be to the m-th sensor, including the propagation channel (i.e., L = 1 P = 1 1 recovered by assuming d and setting . the Green’s function) and the front end ﬁltering , and vm (t) is the additive sensor noise term, assumed as statistically inde- A. Space time covariance matrix pendent of source signals. All signal components can be either complex envelopes with respect to an angular frequency ω0, or To tackle the consistency problem, cross-temporal mo- real valued base-band signals. For a down conversion carrier ments are required to model the spectral spread. To this angular frequency ω0 and a sampling period T , the array purpose, the Space-Time Covariance Matrix (STCM) RST = H response at the discrete time angular frequency ν coincides E xST (n) xST (n) [27] is a natural starting point for with the response to a continuous time angular frequency exploiting the model (6), because it collects all available ω (ν) = ν/T + ω0. second-order information for zero mean, circularly complex The sequences of consecutive P array output samples signals and noise with bounded fourth-order moments, if P > max (Ld + Ls,d), where Ls,d is the correlation length xm (n) = xm (t)|t=nT for n = 1, 2,...,N are collected in a d space time snapshot (STS) of dimension MP × 1 of the dth source2. It is worth to point out that in the same

T T T T scenario the SCM estimate would require the use of a DFT of xST (n) = x1 (n) x2 (n) ··· xM (n) (2) length Pw P > max (Ld + Ls,d), because its bias slowly d −1 where decays as Pw [1], [8], [18]. T In the sequel, the vector of source signals xm (n) = xm (n) xm (n − 1) ··· xm (n − P + 1) . T T T (3) s (n) = s1 (n) ··· sD (n) (10) Each discrete time sensor impulse response hm (θd) (m = 1, 2,...,M, d = 1, 2,...,D) represents the discrete time is assumed as a realization of a circular, wide sense stationary process with zero mean and covariance counterpart of hm (t; θd) and is described by the row vector   S11 ··· S1D hm (θd) = hm,d (0) hm,d (1) ··· hm,d (Lm,d − 1) . H S = E s (n) s (n) =  ·········  (11) (4) H S1D ··· SDD Its duration Lm,d may be inﬁnite, if hm (θd) contains poles H [38]. However, |hm,d (n)| typically quickly decays with n whose blocks Skl = E sk (n) sl (n) are the cross- well under the noise ﬂoor and it makes sense to assume a covariance matrices between each pair of impinging signals.

1 2 Following this formulation, hm (t; θd) may collect the effects of multiple The use of other statistics based on (6), such as biased [22], structured coherent reﬂections originated by the same driving signal sd (t) [37]. [39] or cyclic correlation matrices [29], is deferred to future work. 4 SUBMITTED

Sensor noise is assumed zero mean, circular and inde- However, the wide-band ML and WSF approaches based pendent of source signals, with STCM E v (n) vH (n) = on (6) lead to awkward, huge-dimensional problems that need λvRvv where λv ≥ 0. Then, the array STCM is to be initialized by accurate (within half beam-width or less [2]) prior DOA estimates for successful local convergence. In R = x (n) xH (n) = H SHH + λ R ST E ST ST ST ST v vv (12) particular, the difficulty of measuring the full H(θ) [38] mo- where HST = H (θ1) ··· H (θD) . tivated us to develop approaches based only on the knowledge of the array harmonic response at a discrete set of temporal frequencies. B. Space time signal and noise subspaces H For known Rvv = RvRv , the orthonormal and comple- IV. SPACE TIME MUSIC mentary bases Es for the wide-band signal subspace and Ev In [22], a set of consistent and optimally weighted in for the wide-band noise subspace are classically defined by the WSF sense [6] narrow-band signal subspace bases at the eigen-decomposition (EVD) [36] of the whitened STCM arbitrary frequencies were obtained directly from Es through −1 −H H H an extension of the MUSIC paradigm. This solution, referred Rv RST Rv = EsΛsEs + λvEvEv (13) to as Space-Time MUSIC (ST-MUSIC), bypasses traditional where the diagonal matrix Λs = diag{λ1, . . . , λη} contains filtering, SCM building and EVD stages. In the sequel, the the dominant η eigenvalues in non-increasing order (i.e., λ1 ≥ STCM signal model at frequency ν is generalized to the non λ2 ≥ ... ≥ λη > λv). In particular, the subspace Es has spherical noise case and the ML ST-MUSIC subspace estimate dimension is derived from the solution of an inversion problem from Es. D The space-time array response to a pure unit amplitude, X jνn η ≤ (P + Ld − 1) ≤ D P + max (Ld) − 1 (14) complex sinusoid s (n) = e , impinging from the generic d d=1 angle θ, can be written as and the column span of E lies within the column span of jνn s xST (n) = aST (ν, θ) e (16) HST , so that [6]   where aST (ν, θ), referred to as the space time steering vector C1 (STSV), has size MP × 1 and is defined by the Kronecker −1 H (θ ) ··· H (θ )  .  Es = Rv 1 D  .  (15) product CD 1/2 aST (ν, θ) = a (ν, θ) ⊗ eP (ν) = P E (ν) a (ν, θ) (17) where the blocks Cd for d = 1, 2,...,D have size where a (ν, θ) is the classical M × 1 narrow-band ar- (P + Ld − 1)×η and point out the contributes of each source. ray steering vector at frequency ν [20] and eP (ν) = T 1 e−jν ··· e−j(P −1)ν is a vector of size P ×1 [27], C. DOA identifiability from the STCM by ML and WSF [31]. The last equality in (17) defines the MP ×M orthogonal approaches matrix −1/2 Model (12) is theoretically amenable to the straight ex- E (ν) = P IM ⊗ eP (ν) (18) tension of ML [1]–[3] and WSF [5], [6] narrow-band DOA as a basis for the subspace spanned by any aST (ν, θ). estimators3, assuming for instance the exact knowledge of To put into evidence the relationship between aST (ν, θ) and the matrix impulse response H(θ) for all θ, of the noise H (θ), let us define the unitary matrix of size P + Ld − 1 covariance Rvv up to a positive scalar, and of the exact number h −1/2 i of paths to search for. In addition, the source combination D(ν) = (P + Ld − 1) eP +Ld−1 (ν) D⊥ (ν) which achieves the target RST must be unique [20]. (19) In particular, the rank η of Es must remain smaller than where D⊥ (ν) is the orthogonal complement [36] to MP E for the existence of v, crucial for the unambiguous iden- eP +Ld−1 (ν). H tification of λv and DOAs [6], [20]. By (6), each uncorrelated After posing s (n, ν) = D (ν) s (n) and F,d Ld+P −1 d source contributes to η by a variable quantity ranging from one H (ν, θd) = H (θd) DL +P −1 (ν) = (e.g., for a pure complex sinusoid) [22] up to P + Ld − 1 for d (20) a random source spanning the full array bandwidth. Since for aST (ν, θd) H (ν, θd) D⊥ (ν) D full-band sources η > DP , the number of resolvable wide- the response (6) of the d-th source signal after noise whitening band sources is strictly smaller than M. Further limitations on can be rewritten by (16) as the identifiability may be imposed by array ambiguities and −1 −1 Rv xST (n) = Rv H (ν, θd) sF,d (n, ν) = spatial coherence [20]. −1 1/2 Rv P E (ν) a (ν, θd) H (θd) D⊥ (ν) sF,d (n, ν) . On the contrary, narrow-band sources generate only a few (21) significant eigenvalues and proper techniques [40]–[42] can To describe the effects of noise whitening on the signal already estimate more DOAs than sensors if the source spectra model at frequency ν, let us define the following full QR do not overlap. decomposition (QRD) 3 R−1 (ν) For zero mean circular Gaussian signals and noise the STCM is a sufficient P 1/2R−1E (ν) = A (ν) A (ν) v (22) statistic for DOA and S parameters of model (6). v ⊥ 0 DI CLAUDIO et al.: SPACE TIME MUSIC: CONSISTENT SIGNAL SUBSPACE ESTIMATION FOR WIDE-BAND SENSOR ARRAYS 5 where A (ν) is the M dimensional frequency subspace basis the steering vectors of the fully coherent arrivals must be −1 at ν, A⊥ (ν) is its orthogonal complement and Rv (ν) is a calibrated and Bs (ν) must be analyzed by a multi-dimensional square upper triangular matrix of size M, which defines the (e.g., WSF type) DOA estimator [6], [25] or subjected to following noise whitened narrow-band steering vector of size spatial or frequency smoothing [8], [12], [43]. M × 1 As usual, for unambiguous DOA identification, it is required −1 b (ν, θ) = Rv (ν) a (ν, θ) . (23) that D (ν) < M and that any subset of D (ν) steering vectors b (ν, θ) is linearly independent, at least in a neighborhood of Combining (21), (22) and (23) leads to the decomposition the source DOAs [20]. AH (ν) R−1H (ν, θ ) s (ν, n) = A viable method for consistently recovering Bs (ν) from AH (ν) v d F,d (25) is to apply a η × κ (ν) inversion weight matrix W (ν) to ⊥ (24) b (ν, θd) B12 (ν, θd) the right side of Es to find a M dimensional basis entirely sF,d (ν, n) . 0 B22 (ν, θd) lying within the span of A (ν): The first M rows contain the sum of the desired, rank one EsW (ν) = A (ν) Bs (ν) . (28) signal component at frequency ν, characterized by b (ν, θd), and of a spectral leakage component transmitted by the In particular, W (ν) must cancel the out of band Es M × (P + Ld − 2) matrix B12 (ν, θd). The latter term arises components in the subspace A⊥ (ν) from the finite array aperture and represents the multiple B (ν) C (ν) W (ν) = 0 (29) rank response of a spatially spread, uncalibrated ghost source 22 F 2 [12]. Its strength depends on the spectrum of sd (n) and its to cancel also the spectral leakage in the bin subspace A (ν) spatial spread on the change of a (ν, θ) with frequency, which increases with the fractional bandwidth and the array aperture. B12 (ν) CF 2 (ν) W (ν) = 0 . (30) Finally, the isolated spectral leakage components can be All involved matrices can be rank deficient under certain observed in the subspace A⊥ (ν) through the transfer matrix conditions. In general, it is immediate to show that the signal B22 (ν, θd). subspace can be exactly recovered iff Eq. (24) is immediately generalized to the entire Es (15) as 2 H |B12 (ν) CF 2 (ν) w|2 A (ν) B11 (ν) B12 (ν) CF 1 (ν) 2 < ∞ (31) H Es = (25) |B22 (ν) CF 2 (ν) w| A⊥ (ν) 0 B22 (ν) CF 2 (ν) 2 w where B (ν) = b (ν, θ ) ··· b (ν, θ ) is the M × for any non-zero vector which does not lie in the 11 1 D B (ν) C (ν) D (ν) narrow-band array transfer matrix at frequency ν of intersection of the null-spaces of 12 F 2 and B (ν) C (ν)4 the D (ν) ≤ D sources with non-zero spectra at the same ν, 22 F 2 . Condition (31) just moves to the frequency domain the combined by the matrix CF 1 (ν) of size D (ν) × η. requirement of non-perfect coherency among narrow-band Finally the mixing matrix CF 2 (ν) of size PD sources for the applicability of the spatial-only MUSIC [20] d=1 (P + Ld − 2) × η models the spectral leakage observable through the transfer matrices (i.e., spectral vs. spatial coherency), but cannot hamper the identifiability of spatially coherent sources from B (ν). s B12 (ν) = B12 (ν, θ1) ··· B12 (ν, θD) The structure of (15) reveals that W can cancel out the (26) B22 (ν) = B22 (ν, θ1) ··· B22 (ν, θD) spectral leakage from Es separately for each uncorrelated that mark the departure of the convolutional signal model (25) signal, so Bs (ν) contains linear combinations of all the from the ideal narrow-band one [20]. steering vectors of active sources. Therefore, the limit ST- MUSIC equation [22], [31], A. ST-MUSIC as an Inversion Problem H Ev A (ν) Bs (ν) = 0 , (32) With reference to (25), a consistent narrow-band signal herein obtained by projecting both sides of (28) onto E is subspace estimator must asymptotically recover for N/MP → v a sufficient condition for the consistency of a signal subspace ∞ an M × κ (ν) linearly independent basis B (ν) (with s estimate. κ (ν) ≤ D (ν) < M) within the span of B (ν) and a full 11 DOA estimators starting from approximations of (32) were rank, square subspace weighting matrix C (ν) of size κ (ν), s considered in the past [4], [13], [42]. However, they employed chosen to optimize the statistical accuracy, satisfying the WSF non consistent, SCM based noise subspace formulations, ba- equation sically unable to exploit the source power information and B (ν) C (ν) = B (ν) C (ν) (27) s s 11 11 fruitfully deal with spatially coherent scenarios. for a proper full rank mixing matrix C11 (ν) of size D (ν) × On the contrary, with consistent estimates of Es and Ev κ (ν) [6]. for N/MP → ∞, the sample ST-MUSIC subspace Bˆ s (ν) In this formulation κ (ν) is the number of uncorrelated asymptotically matches the ideal narrow-band signal subspace source signals that are active at ν [42]. As for the SCM case, 4Intersection of these null spaces may not be empty for sources with spectral spatially coherent (multipath) sources with Ld + Ls,d < P nulls within the array bandwidth, but in this case there is not any leakage to have a single rank signature in Bs (ν) and the DOA identifia- cancel. Infinite (31) indicates that a component does exist at frequency ν and bility is subjected to the same limitations [6]. In particular, not elsewhere, so it must be included in B11 (ν). 6 SUBMITTED model [20] and retains the full flexibility of the WSF approach from the sample version of (32) [22] and leads to a generalized in narrowband and broadband scenarios [5], [6], [12] . eigen-problem. A different problem of ST-MUSIC is the unduly atten- uation of some sources of interest by (28), evaluated as A. Sample STCM eigenvectors B11 (ν) CF 1 (ν) W (ν). In particular, harmonic sources made ˆ ˆ up by less than L + P − 1 sinusoids with all frequencies The sample bases Es and Ev are rotated versions of the different from ν or strongly cyclo-stationary sources with cycle true ones in (13) [36] and converge to these for N/MP → ∞ frequency α < N −1 [30] do not satisfy (32) and might be [44]. The (first-order) partitioned rotation matrix is defined by utterly suppressed. In fact, these sources have a rank deficient I + G G covariance block S in (11). Eˆ Eˆ ≈ E E ss ns (35) dd s v s v G I + G A basic ST-MUSIC subspace estimator was presented in sn nn [22], starting from a rank-reducing approach applied to Es, H H where Gns = −Gsn, Gss = −Gss and Gnn = conceptually similar to TOPS [13]. An approximate ST- H −Gnn are random perturbation derivatives with L2 norm of MUSIC for low SNR applications appeared in [31], based O |R | N −1/2 [44]. H ST 2 on the SVD of the weighted A(ν) Es. These estimators The analysis is simplified by recognizing that blocks used biased STCM estimates in finite samples and could not −1/2 I + Gss and I + Gnn are O N approximations to uni- estimate κ (ν) in a statistically sound way. tary matrices [47]. The invariant subspace Ev is intrinsically For these reasons, in the sequel we set up a ML inversion defined up to an arbitrary rotation. The rotation within Eˆ s −1/2 problem of Bs (ν), based on the first order (i.e., O N ) was considered in random matrix theory [48] and covariance perturbation model [6], [12], [36], [44] of the sample counter- shrinkage [32] and is strong for close signal eigenvalues. part of (13) While the influence of this rotation on the accuracy bounds −1 ˆ −H ˆ ˆ ˆH Rv RST Rv = EΛE (33) of DOA estimates is unknown, (32) shows that full rank where linear mixing of Es is unessential for estimator derivation, N since a backward transformation is induced on W (ν) by the 1 X ˆ Rˆ = x (n) xH (n) (34) numerical optimization on the particular realization of Es. In ST N − P + 1 ST ST n=P addition, attempts to take into account Gss may face undesired problems [32]. is the unbiased STCM estimate built from N array output After these observations, effects caused by G and G samples (1). The sample eigenvalues of (33), λˆ = Λˆ (k, k) ss nn k can be mostly neglected. Instead the random entries of G are ordered for k = 1, 2,...,MP in a non-increasing manner. sn are the most relevant for DOA estimation. They are the finite The corresponding orthonormal sample eigenvectors Eˆ (:, k) sample cosines between Eˆ and the true E , calculated as [6], are partitioned into the sample wide-band signal subspace s v [36], [44] Eˆ s = Eˆ (:, 1 : η), of dimension 0 ≤ η < MP , related to the ˆ ρˆsn (p, q) dominant λk, and the complementary sample wide-band noise Gsn (p, q) = (36) λq − λv subspace Eˆ v = Eˆ (:, η + 1 : MP ), related to the smallest MP − η eigenvalues, clustered around λv [22], [31]. for p = 1, 2,...,MP −η and q = 1, 2, . . . , η. In this equation, N The EVD version of (33) is preferable since the statistical 1 P ∗ ˆ ρˆsn (p, q) = N−P +1 yη+p (n) yq (n) is the finite sample identification of a spherical Ev ensures the feasibility of n=P consistent DOA estimation and reduces the parametrization correlation between the uncorrelated signals yη+p (n) and H −1 and the Mean Square Error (MSE) w.r.t. the true STCM [32]. yq (n), where the generic yk (n) = E(:, k) Rv xST (n) is Under the hypotheses made in Sect. III, two distinct STSs the signal extracted by the transformed true k-th eigenvec- −H (2) xST (n1) and xST (n2) cannot be considered as sta- tor Rv E (:, k), considered as a DS beamformer [37] and tistically independent if |n1 − n2| ≤ P . However, using referred to as eigenfilter. 2 −1 independent STSs would require N MP to get a stable Therefore, E {ρˆsn (p, q)} = 0 + O N , while the co- ˆ RST . variances among Gsn entries are affected by the temporal In contrast, theoretical arguments [45] and past experience correlation between yη+p (n) and yq (n) [49]. For stationary [22], [27], [31] show that (34) converges only for N MP signal and noise, the following covariances are computed after to its asymptotic performance, very similar (except for leakage [33], [44] issues) to the one of a set of SCM estimates, using the same E {ρˆ (p, q)ρ ˆ∗ (k, l)} = N P = P sn sn and a DFT length w , despite the much larger number N−P of degrees of freedom of the STCM. The following analysis P N−P +1−τ E y (n) y∗ (n) × (N−P +1)2 η+p q (37) sheds light on this observation. τ=−(N−P ) ∗ o yη+k (n + τ) yl (n + τ) V. ML ST-MUSIC SUBSPACE ESTIMATOR

The present STCM analysis extends the classical one de- E {ρˆsn (p, q)ρ ˆsn (k, l)} = veloped for the SCM [6], [36], [44], [46], points out the N−P P N−P +1−|τ| E y (n) y∗ (n) × (N−P +1)2 η+p q (38) differences and is validated by the simulations in Sect. VII. In τ=−(N−P ) particular, it shows that the ML ST-MUSIC subspace departs ∗ yη+l (n + τ) yl (n + τ)} . DI CLAUDIO et al.: SPACE TIME MUSIC: CONSISTENT SIGNAL SUBSPACE ESTIMATION FOR WIDE-BAND SENSOR ARRAYS 7

Finally, if signals and noise are realizations of zero mean, circularly complex processes for every τ with zero third-order moments, it is possible to approximate (37) and (38) as ∗ −1 E {ρˆsn (p, q)ρ ˆsn (k, l)} ≈ (N − P + 1) δpkδqlλvλq (39)

E {ρˆsn (p, q)ρ ˆsn (k, l)} = 0 (40) independently of the exact distributions of signals and noise [44]. For N MP , the entries of Gsn approach a Gaussian distribution by a Central Limit Theorem argument [6]. Then asymptotically (36) can be approximated as 1/2 Gsn ≈ G0Γ0 (41)

where G0 is a (MP − η) × η random matrix with i.i.d., zero mean circular Gaussian entries of variance (N − P + 1)−1 and 1/2 nh 1/2 1/2 io Γ0 = diag γ0 (1) ··· γ0 (η) (42) −2 where γ0 (k) = λkλv(λk − λv) , exactly like in the narrow- band case using N − P + 1 independent snapshots [6]. Because of the random rotations Gss and Gnn, G0 is not observable, but ﬁltered versions of it enter the analysis H as large random Wigner covariance matrices (e.g., G0 G0). Wigner matrices converge to a common limit distribution with Gaussian off-diagonal entries and a semicircle eigenvalue distribution, regardless reasonable violations of (39) and (40) [51]. This observation well explains the empirical results of [22], [27], [31]. Curiously, the largest deviations are expected for very small γ0 (k) ≈ λv/λk, i.e., at high SNR, and in the presence of strong temporal correlation of source signals.

B. Normalized signal eigenvectors Fig. 2. Top: Sample noise STCM eigenvector beampattern versus wavenumber and angular frequency, normalized to the array bandwidth center for four Perturbed signal eigenvectors in (35) are orthonormal only pass-band, uncorrelated sources, impinging on the ULA used for simulations up to O N −1/2 terms [44]. From (42), these eigenvectors in Sect. VII-B for SNR= 20 dB. Bottom: Average beampattern computed exhibit a non-physical singularity for λk → λv, when sample from the frequency-interpolated least dominant SCM eigenvectors of 64 DFT ˆ bins from the same realization, showing null depth reduction and bias due to unit norm eigenvectors only slip in the estimated Ev. To cir- leakage artifacts. cumvent this problem, in the sequel we will normalize the k-th perturbed signal eigenvector (35) with respect to its expected p L2 norm 1 + cγ0 (k), where c = (MP − η)/(N − P + 1) With a large computational effort, some statistics of (37) is the ratio between the dimension of Ev and the number and (38) might be estimated from the sample yq (n). However, of STSs used in (34) and resembles a parameter defined in a great simplification can be obtained by examining the random matrix theory [48], [52]. properties of STCM eigenfilters. This fixed O N −1 scaling cannot modify the asymptotic In fact, the beampatterns of signal eigenfilters in the fre- subspace performance for N → ∞ [44]. Even if perturbations quency/DOA space exhibit main-lobes pointing at the regions do not likely follow the first order modeling for λk → λv, this of activity of sources and extract convolutive mixtures yq (n) scaling brings out correction terms that taper off the influence (q = 1, 2, . . . , η) of source signals plus noise. of signal eigenvalues down to zero in the same limit and On the contrary, as shown in Fig. 2, which realizes Fig. partially move marginal eigenvectors into Eˆ v. However, the 1 in a computer simulation, noise eigenfilters generally have corrections are not negligible for typical values of N and Γ, beampatterns with very deep broad-band nulls steered toward therefore increasing the robustness to estimation errors of η. the source DOAs [4]. Therefore, they extract a set of noise The scaled perturbation model (35) of Eˆ s, neglecting the dominated signals yη+p (n) (p = 1, 2,...,MP − η) that inessential rotation induced by Gss, becomes are almost white and mutually uncorrelated and additionally ˆ 1/2 1/2 Es ≈ EsΓs + EvG0Γ (43) uncorrelated with any source dominated signal yq (n) (q = 1/2 1/2 1, 2, . . . , η), because (33) removes in a Least Squares (LS) where Γs and Γ are the diagonal square roots [36] of −1 −1 sense [50] the temporal correlation from yη+p (n) for any Γs = [Iη + cΓ0] and Γ = Γ0[Iη + cΓ0] . In addition, τ < P . 0 < cΓ (k, k) < 1 and (41) can be recovered for c → 0. 8 SUBMITTED

C. Signal subspace rank estimation for the STCM Information Theoretic criteria [23], [52] developed for the SCM are questionable for the STCM, because signal and noise From the previous arguments, it is clear that the rank η of Es is not directly related to the number of uncorrelated impinging processes exhibit very different and a priori unknown numbers signals, as in the narrow-band case [23], but it represents of effective observations by (44) and the likelihood form is the number of STCM signal components that exceed the different for each signal distribution. However, the Gaussian log likelihood ratio [23] for inde- noise level λv and carry significant information about source parameters. In essence, the problem is to find a spherical pendent observations essentially relies on the noise eigenvalue noise subspace centered on a consistent O N −1/2 estimate spread, that changes little under the assumption of Sect. V-A, even for many non-Gaussian noise distributions [45]. These of λv. This goal involves the STCM eigenvalue statistics, still affected by the snapshot dependency issue [49]. arguments support the provisional applicability of unmodified Under the same assumptions made in Sect. V-A, the per- existing SCM rank estimation criteria to the STCM. In a ˆ refined model, the dependent observation issues might be turbative model for the sample STCM eigenvalues λk starts N handled by assuming a smaller N¯ < N − P + 1 number ˆ −1 P 2 from the approximation λk ≈ (N − P + 1) |yk (n)| of observations [49] in the relevant noise subspace, but a n=P n o quantitative study is beyond the scope of this work. [36], [44]. It is deduced that λˆ = λ + O N −1 E k k In particular, the AIC [23], which takes into account only n o ˆ −1 for k = 1, 2, . . . , η and E λk = λv + O N for the validation risk [35] of adding a new signal eigenvector, k = η + 1, η + 2,...,MP . performed better, especially at low SNR, in comparison with ˆ ˆ For large N, the covariance between λk and λl, correspond- the non parametric approach of [12], the BIC [23] and the ing to distinct λk 6= λl, as well as the variance of isolated modified AIC of [52], that evidenced clear mismatches of their ˆ signal λk, approach [44], [49] additional parameters. In particular, despite of consistency claims and stable η estimation at high SNR, the BIC exhibited N−P cov λˆ , λˆ ' P N−P +1−|τ| × a disappointing detection threshold at low SNR, about 10 dB k l (N−P +1)2 τ=−(N−P ) (44) higher than the AIC one. Therefore, only the AIC will be n 2 2o E |yk (n)| |yl (n + τ)| − λkλl . included in the simulations of Sect. VII.

In particular, pairs of signal and noise sample eigenvalues D. ML ST-MUSIC inversion are almost uncorrelated. However, (44) indicates that the variance of signal STCM eigenvalues is much higher in the The main issue with earlier versions of ST-MUSIC [22], ˆH dependent snapshot case, but still generally lower than in the [31], based on the SVD of Es A (ν), was the statistical SCM case. instability of the signal subspace weighting, which generalized As regards the sample noise eigenvalue statistics, the con- existing weighted MUSIC schemes [25], [33]. In particular, the ˆ sistency of the classical noise variance estimate [23] empirical estimate Γ of the large Γ led to signal cancellation at high SNR, as in adaptive beamforming [28]. Heuristic MP −η ˆ 1 X regularization of Γ, based on [32], stabilized the subspace λˆ = λˆ (45) v MP − η k estimate at the expense of the sample DOA bias, partially k=η+1 spoiling the goal of consistency. follows from the convergence of (34) to the true STCM under A true ML formulation circumvented these issues by rewrit- the given assumptions on the choice of P [45]. ing the sample version of (28) at frequency ν, after inserting In addition, the robust censoring estimator of η pro- (43), as ensemble variance posed in [12] showed that the sample ˆ 1/2 −1/2 nˆ o EsW (ν) = A (ν) B (ν) + EvG0Γ W (ν) + o N var λη+p; p = 1, 2,...,MP − η of (33) is always close 2 (46) to cλv for Gaussian noise, as in the independent snapshot where B (ν) is the unknown candidate signal subspace basis case [52]. This result confirms that the extracted noise signals at (ν) for Bs (ν) in (27). yη+p (n) behave as white and mutually uncorrelated for any It is assumed that η has been estimated as described in τ of interest, in agreement with the analysis of [45] for Sect.V-C. The true Γ and W (ν) are assumed initially known. more general noise distributions. However, a small increase In the sequel, for conciseness we will set NP = N − P + 1 of this ensemble variance was observed at low SNR and was and drop the reference to ν, e.g., A = A (ν), B = B (ν) and interpreted as a symptom of increasing leakage of temporally W = W (ν). correlated signals into Eˆ v . 1/2 The sample equation error EvG0Γ W of (46) lies in Therefore, by Chebyschev’s inequality, the lower threshold true E √ ˆ the v. One of the instability sources was found in the for signal eigenvalues should be a small multiple of cλv E Eˆ ˆ classical replacement of v by the sample v, that called for above λv to avoid missing the small components originated a more sophisticated route. Projecting (46) onto Eˆ v and Eˆ s from the fading tails of the impulse responses hm (θd) [22]. leads to It follows that any optimal value of η shrinks with the SNR ˆ H 1/2 ˆ H −1/2 and this phenomenon may impact on the DOA identifiability Ev EvG0Γ W = Ev AB + o N (47) by (32), since a reduced dimension W (ν) may not consis- 1/2 H 1/2 ˆ H −1/2 tently recover the full span of Bs (ν) anymore. Iη − Γ G0 G0Γ W = Es AB + o N (48) DI CLAUDIO et al.: SPACE TIME MUSIC: CONSISTENT SIGNAL SUBSPACE ESTIMATION FOR WIDE-BAND SENSOR ARRAYS 9 and w.r.t. W, leading to the asymptotically unbiased estimate ˆ H 1/2 1/2 H 1/2 −1/2 ˆ ˆH Es EvG0Γ W = Γ G0 G0Γ W + o N . W = ΦEs AB (53) (49) 2 −1 2−1 Equations (47) and (49) together characterize a row rotation where Φ = Iη − 2cΓ + c + cηNP Γ (Iη − cΓ) is a 1/2 of EvG0Γ W, while (48) can be used to estimate W. After real-valued, diagonal matrix of size η. 1/2 (41), G0Γ W can be modeled without loss of generality as The candidate solution for B can be found from (52) a stack of MP − η independent, zero mean, Gaussian circular through the Generalized SVD (GSVD) [36], which leads in −1 H a square root fashion to the decompositions observations of dimension M with covariance NP W ΓW , characterized by the following negative log likelihood for H H ˆ ˆ H 2 H ˆ 2 ˆ 2 H given W and Γ Πε = A EvEv A+c A EsΦΓ ΦEs A = FΣεF (54) H ˆ ˆH 2 H ϕ (M) = M (MP − η) ln (π) + Πs = A EsΦΓΦEs A = FΣsF (55) (MP − η) ln det N −1WH ΓW + P (50) where Σ2 and Σ2 are diagonal square matrices of size M with h (0) H −1i s ε NP trace Πε W ΓW 2 2 non-negative diagonal entries, linked by Σε + Σs = IM , and F is a complex valued square matrix of size M. By posing where −H (0) B = F Π = BH AH Eˆ EˆH AB+ [36], (52) can be simplified as ε v v . H 1/2 H H 1/2 −1 W Γ G0 G0ΓG0 G0Γ W ϕ (M) = M (MP − η) ln πNP + (MP − η) ln det Σ2 + cMtrace (Γ) + (56) (0) s The last term of Πε can be replaced by its expected value 2 −2 NP trace ΣεΣs . under the Gaussian i.i.d. approximation on G0, leading to Not every column of B is acceptable as a basis vector for the (0) H H ˆ ˆ H Πε = B A EvEv AB+ signal subspace, since ϕ (M) has small contributes only along H 2 2 −1 (51) W c Γ + cΓ · NP trace (Γ) W . the κ columns of B characterized by very small generalized 2 2 eigenvalues µk = Σε (k, k) Σs (k, k) ' c. H −1 The term W ΓW in (50) can be interpreted as the In fact, plugging (43) into (54) and (55) and taking the −1 high resolution Capon estimate [19] of the power spectral expected value over G0, leads to the following O N density (PSD) at frequency ν of a set of signals characterized approximations −1 by the covariance Γ and the steering vector matrix W. As a n o H ˆ 2 2 ˆH consequence, signals components at frequencies different from E {Πε}' IM − E A Es Iη − c ΦΓ Φ Es A ν, such as spectral leakage products, are suppressed as in MV H H (57) ' A EscΓ (Iη + cΓ) Es A+ beamforming [28]. The same term in (51) compensates for the 1 − N −1trace Γ − c2Γ3Φ2 AH E EH A ˆ P v v inclusion of marginal signal eigenvalues close to λv. −2 2 H 2 H H The O N term c W Γ W in (51) cannot influence E {Πs}' A EsΓ (Iη + cΓ) Es A+ −1 2 2 H H (58) the asymptotic performance, but it is the key for regularizing NP trace Γ Φ A EvEv A (0) (50) at any SNR by filling the numerical rank of Πε and where some non-negligible terms from the Taylor series ex- compensating for any rotation of the sample W. Its exact H pansions of (53), (54) and (55) w.r.t. c are retained for clarity scaling mildly depends on the assumed distribution of G0 G0, and future use. but the given calculus for the Gaussian case is sufficient for Thus, on the average, for N → ∞ the generalized eigen- regularization purposes. equation corresponding to (54) and (55) [36] Plugging (51) into (50) and simplifying leads to −1 E {Πε} B (:, k) ' µk E {Πs} B (:, k) (59) ϕ (M) = M (MP − η) ln πNP + H (MP − η) ln det W ΓW + cMtrace (Γ) + NP × is asymptotically satisfied by an eigenvalue µk = c of multi- h H H H 2 H 2 H −1i trace B A Eˆ vEˆ AB + c W Γ W W ΓW . plicity κ. The corresponding κ eigenvectors, collected in the v H H (52) M × κ matrix Bs, asymptotically satisfy A EvEv ABs ' 0 There is some freedom in choosing an estimate Wˆ of W and therefore (32). from (48). Early weighted forms of MUSIC DOA estimators In addition, the null-space of E {Πε − cΠs} is independent [25], [33] obtained null spectra resembling the last term of of N for sufficiently large samples, demonstrating that the ˆ ˆ ˆ H span of the sample Bs (i.e., Bs) provides an asymptotically (50) for the implicit choice W = Es AB. However, errors in Wˆ may destroy the Capon spectral estimate of (52) [28]. unbiased ST-MUSIC signal subspace estimate up to at least −1 Therefore, we sought for linear estimates of W, independent O N terms. of G , that minimize both the residual error of (48) and the This claim is not evidently shared by earlier ST-MUSIC 0 −1 ˆ estimators [22], [31] that retain O N signal subspace variance of W, according to (43). In particular, the form of n o H ˆ ˆH the random term of (43) suggested to minimize the expected components in E A EvEv A , related to the source spec- value over G0 of the weighted LS functional tra. In fact, the different asymptotic bias at various frequencies 2 is another source of DOA estimate instability at high SNR in −1/2 hˆ H 1/2 H 1/2 i Γ Es AB − Iη − Γ G0 G0Γ W wide-band applications without a regularized Γˆ. F 10 SUBMITTED

However, beside these ﬁnite sample effects, the asymptotic The estimates of κ abruptly shrink at very low SNR, variances of the various ST-MUSIC subspaces are identical, as as in the sample SCM case [52]. However, the eigenvec- for the narrow-band MUSIC DOA estimator [25], [33], [48]. tors discarded by (61) and (62) are certainly dominated by For the leakage components to be discarded, in- whitened leakage residuals, are numerically wobbly and must 2 ˆ stead, Σε (k, k) is generally close to one according be excluded from Bs. to (32), so µk is comparable to the Capon PSD On the other hand, (60) involves a small ratio between the −1 h H H H i number of observations MP − η and κ < M and the risk for B(:, k) A EˆsΦΓΦEˆ AB (:, k) , which is generally s DOA estimation of including marginal µ s originated by false much greater than c [52]. However, in rare scenarios, leakage k alarms is low. Therefore, the consistency claims of BIC for residuals might generate some µ of order c, interpreted as k (MP − η)/κ → ∞ are weak and the slightly more permissive extremely weak ghost sources, that have a vanishing impact AIC (61) may alleviate the κ shrinkage issue at low SNR and for N → ∞ and can be therefore harmlessly included in Bˆ . s was adopted in simulations. Finally, the M − κ columns of B, corresponding to the discarded µk c, are collected in the matrix Bˆε. F. Optimal ST-MUSIC Subspace Weighting Optimal DOA estimation in the WSF framework [6], E. ST-MUSIC Rank Selection by AIC and BIC [25] entails the calculus at each frequency of interest of n o Due to the high spectral variability of wide-band sources, ˆ ˆ H the spatial perturbation covariance Ξε = E BsBs − the signal subspace rank κ generally changes with frequency n o n o n o ˆ ˆ H ˆ and should be estimated from data. The heuristic rules based E Bs E Bs , where E Bs = Bs from (59) and of the H ˆ optimal subspace weighting matrix C of size κ × κ, which on the singular value magnitude of A(ν) Es, used in earlier sn o ST-MUSIC approaches [22], caused ambiguities. whitens the perturbation of Bs onto E Bˆ ε = Bε. On the contrary, the proposed ML formulation admits the This task requires the O N −1/2 perturbative analysis of use of Information Theoretic Criteria [23], [34], [35] for (59), adapted and simpliﬁed from [36], [44], [53]. The eigen- estimating κ. To this purpose, sample µks are ordered in a non vector perturbation B˙ s is herein decomposed for convenience decreasing manner5 and a set of nested models with reduced as rank K is compared for 0 ≤ K < M. The log-likelihood of B˙ s = Bˆ s − Bs = BsYs + BεYε each model is written as where Iκ + Ys describes a random rotation of Bs, since all ϕ (K) = K (MP − η) ln πN −1 + P signal eigenvalues µk → c asymptotically. (MP − η) ln det Σ2 (1 : K, 1 : K) + cKtrace (Γ) + s Differentiating (59), Yε asymptotically leads to N trace Σ2 (1 : K, 1 : K) Σ−2 (1 : K, 1 : K) P v s (60) − Π˙ ε − cΠ˙ s Bs ' E {Πε − cΠs} BεYε (63) where, by a continuity argument, ϕ (0) = 0. The estimates of κ by AIC and BIC are readily [23], [34], [35] obtained as where, after tedious calculus using (43), (53), (54), (55) ˙ H 1/2 H H κˆ = arg min {2ϕ (K) + 2K (2M − K + 1)} (61) Πε ' Πε − E {Πε}' A Es [−Iη + 0.5cΓ] Γ G0 Ev A AIC H 1/2 H 0≤K≤M−1 +A EvG0Γ [−Iη + 0.5cΓ] Es A (64) ˙ H 3/2 H H κˆBIC = arg min {2ϕ (K) + K (2M − K + 1) ln (MP − η)} Πs ' Πs − E {Πs}' A EsΓ G0 Ev A 0≤K≤M−1 H 3/2 H (65) (62) +A EvG0Γ Es A . since the number of free real parameters for the model order Inserting (57), (58), (64) and (65) into (63) and taking into 2 H K is M for the preliminary whitening GSVD transformation account from (59) that Ev ABs = 0, we get (i.e., an unessential constant overhead), plus K (2M − K) for H 1/2 H ˆ A EvG0Γ (Iη + 0.5cΓ) Es ABs ' the rank K subspace Bs [36], plus K for the generalized −1 2 H H (66) eigenvalues6. 1 − NP trace Γ + cΦΓ Φ A EvEv ABεYε .

Either (61) and (62) implicitly define an environment de- Taking expectations on both sides yields E {Yε} = 0, pendent upper threshold Tµ > c for a valid µk. Their detailed because E {G0} = 0, confirming the absence of asymptotic analysis is complicated by the statistical interaction with the bias. In addition, right multiplying both sides by the unknown prior estimation of η and is left out of the scope of this paper. Cs, neglecting terms independent of ν, using (58) and solving However, since c 1 and µk c for leakage components, for BεYεCs asymptotically yields (61) and (62) furnished reliable and almost indistinguishable H H H H H † detection results in simulation for both κ and DOA estima- Ξε ∝ BsE YεCsCs Yε Bs = A EvEv A × −1 H 2 tion. This result further supports the findings and the basic NP trace CsCs E Σs (1 : κ, 1 : κ) . assumptions made in the STCM analysis. (67) 2 −1 Since from (59) E Σs (1 : κ, 1 : κ) ' (1 + c) Iκ, min- H 2 5 2 −1 2 2 imization of trace C C E Σ (1 : κ, 1 : κ) , subject to The quantity ln Σs (k, k)+c Σε (k, k) Σs (k, k) increases monoton- s s s 2 H ically with µk and Σε (k, k). det CsCs = 1, asymptotically yields [6] 6 2 The columns of B are orthonormal after the whitening and µk, Σε (k, k) 2 and Σs (k, k) are all linked one-to-one. Cs = Iκ (68) DI CLAUDIO et al.: SPACE TIME MUSIC: CONSISTENT SIGNAL SUBSPACE ESTIMATION FOR WIDE-BAND SENSOR ARRAYS 11 or any other unitary matrix. From (59) a consistent estimate In contrast, with reference to (25), the energy from adja- H H † ˆ −2 ˆ H of A EvEv A is BεΣκ Bε , where cent frequencies in the span of B11 (ν) (i.e., the component † 2 2 B11 (ν) B11 (ν) B12 (ν) CF 2 (ν)) is retained by the SCM and Σκ = (1 + c) Σε (κ + 1 : M, κ + 1 : M) − cIM−κ . canceled by the ML ST-MUSIC, which may give to Fourier methods a detection advantage at low SNR with a coarse Such an error covariance matrix confined within the Bˆ ε subspace of dimension M − κ can cause numerical troubles frequency binning. to WSF estimators especially during the coarse DOA ini- The overall statistical efficiency of the ML ST-MUSIC tialization. So the sample Ξˆ ε can be completed by adding subspace, which, as the SCM, is a reduced statistic w.r.t. a rather arbitrary Hermitian matrix spanning the orthogonal the STCM, remains an open question. However, working in complement Qε of Bˆ ε. In particular, the choice an ideal narrowband scenario (i.e., after posing A = IM ), the weightings of the ML ST-MUSIC and of the optimal ˆ ˆ −2 ˆ H H Ξε = κ BεΣκ Bε + ζεQεQε (69) narrowband subspaces [6] are proportional. The STCM allows more flexible strategies in non-stationary −1 ˆ −2 ˆ H where ζε = (M − κ) trace BεΣκ Bε , effectively mini- environments. For instance, a STCM can be built from many disjoint blocks of few STSs each to locate intermittent, elusive mized the distance Ξˆ ε − κζεIM in most cases with ζε ≈ 1. F sources without sacrificing the spectral resolution and the ˆ However, in the presence of strong leakage Ξε can still be ill- effective SNR as in the binning approach [18]. conditioned, hampering for instance the statistical efficiency of common focusing schemes [21]. VI.COMPUTATIONAL ANALYSIS G. DOA estimation from the ML ST-MUSIC signal subspace The ML ST-MUSIC computing cost is largely dominated by the building and the EVD of the STCM. In particular, the Given the associate Ξˆ ε (ν) and Cs (ν), the sample ML STCM computation requires a bulk of M 2NP multiply and ST-MUSIC subspace Bˆ s (ν) at frequency ν can replace the SCM counterpart in any subspace based narrow-band DOA accumulate operations using the apparatus developed for fast estimator. For instance, the optimal WSF estimator minimizes AR identification [50]. The dominant EVD cost of the sample STCM is about 6M 3P 3 complex flops. Viable alternatives over the DOA parameter set Θ = {θ1,..., θD} the functional include fast Lanczos algorithms [36]. 2 − 1 h i 3 2 ˆ 2 ˆ This effort must be compared to the about 6M P +NM ΘWSF = arg min Ξb ε(ν) B (Θ, ν) C − Bs (ν) Cs (ν) w Θ,C F flops required by the SCM approach for the same tasks. To (70) give an idea, in simulations using MATLAB, the full WAVES where B (Θ, ν) is the tentative steering matrix for B11 (ν) in using ML ST-MUSIC ran in about 1.0 s, while the SCM based (25), Cs (ν) = Iκ from (68) and C is an unknown full rank WAVES in about 25 ms for N = 6400, M = 8, P = Pw = 64 mixing matrix [6]. and 33 analyzed frequencies on a PC machine featuring an The extension of (70) to multiple frequencies is straight- Intel Core i7-6700K processor running at 4.2 GHz and 32 forward [5]. It allows consistent and asymptotically efficient GB RAM. However, in most real-time settings, the marginal DOA estimation in a Gaussian scenario at ν, thanks to the cost of STCM processing is comparable with the overhead of ML formulation of the ST-MUSIC subspace and of the WSF coarse DOA initialization techniques using high performance minimum DOA variance property [6]. Consistency holds for wide-band beamformers [15], [16], [54], orthogonal matching any fourth-order bounded scenario in the absence of other pursuit for WSF [5], [6], or refined focusing schemes [10], model mismatches [12], [24], [26]. [12], [21], [54]. The estimator (70) analyzes spectral slices of the STCM [19] with a very narrow effective bandwidth around ν, comparable to the width of a null of (50), virtually extrapolating the VII.COMPUTER SIMULATIONS STCM and eliminating the effects of steering vector changes The performance of ST-MUSIC was assessed by Monte- with frequency, beside spectral leakage. Carlo time domain simulations7 of a ULA with M = 8 omni- Following this argument, working on P equi-spaced fre- directional sensors, equi-spaced by 0.5 wavelengths at the quencies may not optimally exploit the source information by center frequency ν . The sensor bandwidth ranged from 0.6ν ML ST-MUSIC, since some strong narrow-band components 0 0 to 1.4ν . The background Gaussian noise was temporally and might be missed. In particular, (15) indicates that an optimal 0 spatially white (R = I ) and the source SNR was referred ST-MUSIC would require the coherent analysis of at least vv MP to each array element. P + L − 1 optimally selected points of the full z plane (i.e., d For each SNR value, 1000 independent Monte Carlo trials by using a complex ν) [41], [42]. were run, collecting N = 6400 sensor snapshots sampled at In addition, since the consistency of signal subspace fitting T = (0.8ν )−1 s and using P = 64 (N = 6337) for the does not require the exact specification of Ξ (ν) and C (ν) 0 P ε s STCM and an un-windowed DFT of P = P points over [6], [12], the O N −1/2 estimates (68) and (69) are adequate. w However, obtaining robustness to steering vector mismatches 7It is worth noting that inconsistency effects cannot be observed if wide- at ν may require different choices of (68) and (69) [12], [24], band array outputs are simulated in the frequency domain by a set of [26]. independent narrow-band signals. 12 SUBMITTED

Fig. 3. Plot of the DOA sample standard deviation versus the SNR for the Fig. 4. Plot of the DOA sample bias versus the SNR for the source impinging ◦ ◦ source impinging from 37 , observed at the single frequency ν0, obtained from 37 , observed at the single frequency ν0, obtained by MODE applied by MODE [25] applied to different signal subspace estimates, compared with to different signal subspace estimates. the narrow-band SCM square root CRB for Ns = 100.

around 5 dB. The DFT-generated subspace was clearly non- Ns = 100 non overlapping, consecutive data segments for consistent. The ML ST-MUSIC extracted more information SCM based approaches8 [8], [12], [25]. than the classical SCM subspace for Ns = 100, obtaining Four sources emitting various kinds of signals were placed ◦ ◦ ◦ ◦ lower DOA variance, coupled with negligible and similar bias. at the azimuth angles 8 , 13 , 33 and 37 , referred to the These results support the consistency and the superior accuracy array broadside. Performance comparison employed state of potential of STCM and ML ST-MUSIC and can be directly the art WSF type DOA estimators, namely MODE [25] and extended to multi-frequency WSF estimators [5]. WAVES [12] followed by MODE, in turn fed by sets of signal subspaces drawn by ML ST-MUSIC and by SCM, optimally weighted for finite sample errors according to [6], [12], B. Wide band focusing (68) and (69). Since DOA variance and bias have different The test ran in the same scenario, but focusing the center dominant causes (finite sample errors and model mismatches, frequencies of the 33 bins in the band (0.8ν0, 1.2ν0) onto respectively [21]), they were separately analyzed. ν0 using WAVES [12], followed by MODE. In this case, DOA consistency was basically hampered by the focusing A. Single frequency analysis bias [13], [21] and the test emphasized the quality of the This experiment tested the consistency and the basic accu- information combination across frequencies and the robust- racy of the subspace estimates. Sources radiated equi-powered ness to model errors. Unitary focusing matrices were used, and uncorrelated wide-band Gaussian noise, filtered in the assuming that preliminary beamforming estimated two DOA clusters centered at 10.50◦ and 35◦ with focusing sectors as band (0.725ν0, 1.275ν0) (see Fig. 2). MODE (i.e., a root version of (70)) was applied in turn to the ML ST-MUSIC in [8], [12]. [8]. However, the number of focusing angles was ◦ ◦ ◦ ◦ ◦ ◦ subspace estimate at ν , to the corresponding SCM estimate increased to twelve (6.7 , 8.6 , 10.50 , 12.3 , 14.1 , 31 , 0 ◦ ◦ ◦ ◦ ◦ ◦ at bin 0 and to the reference subspace drawn from a classical 32.3 , 33.6 , 35 , 36.3 , 37.6 and 39 ) for better accuracy of array interpolation. In fact, the unitary Procrustes DOA narrow-band SCM [20] built with Ns = 100 independent snapshots, whose Cramer Rao bound (CRB) for DOA was interpolation problem [8], [9], [36] over the sector ensemble S yields a focusing matrix equal to the orthogonal polar factor of also calculated. For the STCM, η = 200 was selected from R H the median of the BIC estimates at the highest SNR of 40 dB. b (ν0, θ) b(ν, θ) dθ. The original angle choice of [8] did θ∈S The ST-MUSIC subspace rank κ was selected online by AIC not adequately cover the effective sector rank and approximate (61). The signal subspace rank of the sample SCM and the this integral [55]. number of sources searched by MODE were instead fixed at The focusing accuracy was still unsatisfactory for our four. purposes at high SNR and forced us to set the WAVES DOA standard deviation and bias are displayed in Figs. 3 rank to four. As an additional measure, a cascaded, non- ◦ and 4 versus the SNR for the source impinging from 37 , the unitary sector interpolation matrix [56], unique for all bins, most critical one for CRB and steering vector change with remapped the principal singular vector of each set formed by frequency. For all algorithms, the low SNR threshold was at the 33 focused steering vectors at the same DOA onto the

8 corresponding ULA steering vector at ν0 [57]. This heuristic Setting Pw = P allowed the use the same focusing matrices in Sects. VII-B and VII-D. Due to the previous discussion, these choices may not be correction substantially reduced the bias and still satisﬁed optimal for each estimator and scenario. Fisher optimality criteria [8]. DI CLAUDIO et al.: SPACE TIME MUSIC: CONSISTENT SIGNAL SUBSPACE ESTIMATION FOR WIDE-BAND SENSOR ARRAYS 13

Fig. 5. Plot of the DOA sample standard deviation versus the SNR for ◦ Fig. 7. Plot of the DOA sample standard deviation versus the SNR for the source impinging from 37 , obtained by WAVES [12], focused on 33 ◦ bin frequencies, followed by MODE applied to different signal subspace the coherent source impinging from 37 , obtained by WAVES, focused on estimates, and compared with the narrow-band SCM square root CRB and 33 bin frequencies, followed by MODE+WSF, applied to different signal subspace estimates and compared with the narrow-band SCM square root the WAVES PSM bound computed for Ns = 100. CRB computed for Ns = 100.

high SNR, further complicated by the ghost source detection when using AIC, and was also heavily biased at every SNR. The DOA variance of the WAVES with ML ST-MUSIC subspaces appeared slightly degraded at extreme SNRs w.r.t. the single frequency case. The most likely causes are the focusing errors and the Fisher information loss [8] due to the suboptimal averaging of Ξˆ ε (ν) across the pass-band, performed by the chosen ﬁxed focusing scheme [12].

C. Wide Band Coherent Sources The test was repeated with similar results after replacing the sources at 13◦ and 37◦ with delayed replicas of those at 8◦ and 33◦ with delays 0.45T and 0.35T respectively [22]. In these trials, there was not any obvious choice for η and Fig. 6. Plot of the DOA sample bias versus the SNR for the source impinging the SCM rank, that were selected by AIC. MODE had local from 37◦ for the various subspace estimates. convergence issues and its DOA estimates were refined by WSF Newton iterations. Results are shown in Figs. 7 and 8. The PSB was not shown since it is ambiguous for coherent The CRB and the SCM based WAVES bound under the arrivals, depending on the focusing strategy. perfect subspace mapping (PSM) assumption [9], [12] were computed for N = 100 independent snapshots drawn from s D. Wide Band Autoregressive Sources each DFT bin. Moreover, beside the cases of fixed selection for η = 200 and the SCM rank, two further experiments were The previous test was repeated after replacing the white included, where these quantities were all estimated online by sources with independent 16-th order autoregressive (AR) AIC [23]. sources, whose models were drawn from live recordings at an 3 Sample results were displayed in Figs. 5 and 6 for the airport, assuming ν0 = 2π × 10 rad/s. The source correlation source at 37◦. The standard deviation plot of Fig. 5 revealed length ranged from 30T to 50T . The SNR was referred to the a slight advantage of SCM over ML ST-MUSIC subspaces driving Gaussian noise variance, equal for all sources. The at low SNR, with a threshold for reliable detection slightly STCM rank estimated by BIC at high SNR was η = 199 in higher for AIC driven rank estimates. The difference appears this environment. essentially due to a slightly higher (not removed) outlier rate Sample results were displayed in Figs. 9 and 10 for the ◦ of ML ST-MUSIC at extremely low SNR, because the AIC source at 37 and confirmed previous findings. detected rank κ shrunk at several frequencies in some trials. Instead, the overall insensitivity of ML ST-MUSIC to the AIC VIII.CONCLUSION selection of η was remarkable, as expected. A consistent signal subspace estimator at arbitrary frequen- The SCM based WAVES exhibited the usual breakdown at cies (ML ST-MUSIC) has been developed by exploiting the 14 SUBMITTED

finite rank representation of wide-band sources in the signal subspace of the STCM through a ML inverse fitting, based on the MUSIC paradigm. AIC and BIC estimators of the number of sources active at each frequency of interest have been naturally derived in this framework. The ML ST-MUSIC approach radically circumvents the frequency spread issues of classical binning and is theoretically supported by the asymptotic nulling of the spectral leakage, identified as the source of the statistical inconsistency. Optimal weighting for finite sample errors allows the use of ML ST-MUSIC subspaces in any narrow- or wide-band subspace fitting DOA estimator. Simulations support the consistency, the superior accuracy (evident at high SNR) and the robustness to mismatches (evident in automatic model selection) of DOA estimation derived from ML ST-MUSIC subspaces w.r.t. the SCM based counterparts. Fig. 8. Plot of the DOA sample bias versus the SNR for the coherent source Finally, the ML ST-MUSIC derivation raises the delicate impinging from 37◦ for the various subspace estimates. question if binning approaches are theoretically justified even in narrow-band scenarios. In fact, since it is always Ld > 1 in physical arrays, a STCM with low order P should be used in order to guarantee consistency.

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